The ideas presented in this paper can readily be extended to
smoothing, i.e., characterizing the conditional distribution
of past states, given all available measurements. In this context,
one can first estimate a (non-singular) Markov transition kernel
between observation times via an upper-triangular KR rearrangement,
and then use this approximation to propagate the filtering ensemble
backwards in time. The resulting scheme can be interpreted as a Monte
Carlo approximation of the transport map backward-smoothing algorithm
introduced in [1]}.
Relatedly, it would be useful to extend the present framework to problems of sequential joint
parameter and state estimation.
